On a time fractional diffusion with nonlocal in time conditions

Tuấn, Nguyễn Huy; Triet, Nguyen Anh; Luc, Nguyen Hoang; Phương, Nguyễn Duy

doi:10.1186/s13662-021-03365-1

Cited by 5 publications

(6 citation statements)

References 36 publications

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“…Using Proposition 1, by virtue of estimates (12) and equality E ρ (0) = 1, we arrive at (see [20], p. 47).…”

Section: Preliminariesmentioning

confidence: 93%

“…As we know, in most models described by differential (and pseudodifferential, see e.g., [10]) equations the initial condition is used. However, in practice, some other models have to use nonlocal conditions, for example, including integrals over time intervals (see, e.g., [11] for reaction-diffusion equations or [12] for fractional equations), or connecting the solution at different times, for instance at the initial time and at the terminal time (see, e.g., [13,14]).…”

Section: Introductionmentioning

confidence: 99%

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On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Ashurov

Fayziev

2022

Fractal Fract

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The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

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“…Using Proposition 1, by virtue of estimates (12) and equality E ρ (0) = 1, we arrive at (see [20], p. 47).…”

Section: Preliminariesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Ashurov

Fayziev

2022

Fractal Fract

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“…Other classes of motivational models for NICs have traditionally been integro-differential and dynamic inclusions on time scales [135][136][137][138], along with delay differential equations and reaction-diffusion systems [139]. While we were already mentioning fractional diffusion problems as one of the key motivations in this field, it is also worthwhile to point out that fractional mathematical models with NICs have been a subject of interest [140] where one of the tools for the analysis of their well-posedness relies on Mittag-Leffler functions, traditionally useful in nonlocal models [141,142].…”

Section: Nonlocality In Timementioning

confidence: 99%

“…In other areas, the interest to nonlocal integral initial conditions and their modified versions is also motivated by mathematical models based on fractional differential equations and inclusions, as well as by problems involving dynamic control with state-dependent requirements [191,192]. They include also stochastic and nonlinear formulations, as well as problems with terminal time conditions, arising naturally in various applications [142,[193][194][195]. Now, we would like to demonstrate the application of the presented results on an example of the problem with three-point nonlocal condition and compare the constraints on nonlocal parameters α 1 , α 2 ∈ R obtained with help of Theorems 2 and 3 against the previously known sufficient condition stated by (5).…”

Section: Lemma 3 All Zeros Of P(u) Satisfy the Double Estimate Max{vmentioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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“…However, there are also processes where we have to use non-local conditions, for example, the integral over time intervals (see, e.g. [8] for reaction diffusion equations or [9] for fractional equations), or connection of solution values at different times, for example, at the initial time and at the final time (see, e.g. [10] - [11]).…”

Section: Introductionmentioning

confidence: 99%

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

Ashurov¹,

Fayziev²

2022

Preprint

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Let A be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space H. The inverse problems of determining the right-hand side of the equation and the function ϕ in the non-local boundary value problemOperator D t on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems u(ξ 1 ) = V is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution u(t) was violated, while when solving the inverse problem for the same values of α, the solution u(t) became unique.

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On a time fractional diffusion with nonlocal in time conditions

Cited by 5 publications

References 36 publications

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

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